Some Nonlinear Elliptic Problems in Unbounded Domains

Abstract

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem $$ (P)\quad \left\{ \begin{array}{*{20}rc} -\Delta u + a(x)u = |u|^{p - 2} u& {\text{in }}\Omega ,\\ u \in H_0^1 (\Omega ), \end{array} \right. $$ where $$\Omega \subseteq \mathbb{R}^N $$ , N ≥ 3, is an unbounded smooth domain, a(x) is a smooth real function defined on Ω, such that $$a(x)\xrightarrow[{|x| \to + \infty }]{}a_\infty > 0,\; p \in \left( {2,\frac{{2N}}{{N - 2}}} \right)$$ , is considered and existence and multiplicity results are given under various assumptions on Ω.